A quantum register stores quantum states including superposition states. However, super-position states are very fragile because coupling to the environment causes dephasing, i. e.
the loss of the known phase relationship between the quantum states of the qubits and the microwave field. Their resulting evolution from pure quantum states to a statistical mixture of states is called decoherence and sets the upper limit as to how long quantum information can be stored and processed.
2.4.1 Crosstalk
Our addressing technique of the quantum register requires a magnetic field gradient so that two qubits with a separation of ∆xwill inevitably acquire a phase of ∆ϕ(t) = ∆xω0t between the states |0i and |1i within a time t. Since this phase difference is known and can be accounted for in possible gate operations, it does not cause decoherence of the quantum register.
A second effect changes the phase relation between the register qubits in a known and predictable way. The application of a microwave pulse at the resonance frequency of one qubit induces a coherent phase shift ∆ϕon adjacent atoms. It is caused by non-resonant interaction with the detuned microwave pulse and is not negligible even if the separation of the qubits is large enough to prevent measurable population transfer. In the far-off resonant limitδΩR, the phase shift can be approximated [51]:
∆ϕ= Z ∞
−∞
ΩR(t)2
2δ dt. (2.13)
If a microwaveπ pulse with the Gaussian pulse shape ΩR(t) of Equation (2.5), is applied with a detuning of δ=ω0∆x, Equation (2.13) yields:
∆ϕ= π32
4στδ. (2.14)
The general case requires the numerical integration of the optical Bloch equations (1.18).
Figure 2.13 (b) shows the resulting phase shift as a function of the position offset for the addressing spectrum of Figure 2.8 (c). For illustration and comparison, the addressing spectrum is presented again in Figure 2.13 (a), where the solid line indicates the calculated
2.4 Coherence properties 69
Figure 2.13: Coherent phase shift induced by non-resonant interaction with a microwave pulse. a) Single atom addressing spectrum of Figure 2.8 (c). The solid line shows the population transfer calculated from a numerical integration of the optical Bloch equations. b) Coherent phase shift inferred from the same calculation, for the two initial conditions u0 = (0,1,0) andu0= (1,0,0). In the shaded area the phase shift between states |0i and |1i cannot be properly defined because significant population transfer occurs. At a position offset corresponding to the addressing resolution of 2.5 µm, the phase shift is ∆ϕ= 0.2π.
population transfer.
The phase shift plotted in Figure 2.13 (b) is inferred from the same calculation, with Bloch vectors initialized in stateu0= (0,1,0) andu0 = (1,0,0). In the shaded area around the atomic resonance, the phase cannot be properly defined because significant population transfer occurs. For large detuning, the phase shift is independent of the initial state u0 and exhibits the 1/δ−relation of Equation (2.14). A second atom separated by the addressing resolution ofσadd= 2.5 µm from the addressed atom acquires a phase shift of
∆ϕ= 0.2·π, independent ofu0.
Since this coherent phase shift is known, it can be taken into account in future gate operations. If desired, it could also be reversed for this specific qubit by an appropriate microwave pulse with opposite detuning.
2.4.2 Optical Bloch equations with damping
Our following investigations show that the relevant timescales for the decoherence of our quantum register are set by dephasing mechanisms which cause the register qubits to lose
their known phase relation. For a formal description of spin relaxation and dephasing within the Bloch vector model, damping terms are added to the optical Bloch equations for an ensemble of atoms [51]:
hu˙i = δhvi −hui T2
hv˙i = −δhui+ ΩRhwi − hvi T2
(2.15) hw˙i = −ΩRhvi −hwi −wst
T1
,
whereh. . .idenotes the ensemble average. The population wdecays to a stationary value wst at the rateT1, which is called longitudinal relaxation time. The transverse dephasing timeT2 can be divided into two different contributions:
1 T2 = 1
T20 + 1
T2∗ . (2.16)
The homogeneous decay timeT20governs the dephasing due to incoherent interactions that affect all atoms in the same way. The inhomogeneous dephasing time T2∗ measures the transverse lifetime due to inhomogeneous effects which assign each atom its own individual resonance frequency and cause macroscopic polarization damping of the ensemble. We will see in the following sections that there is an essential difference between these two types of dephasing. While inhomogeneous dephasing is usually reversible by spin echo pulses, homogeneous dephasing, e. g. due to random magnetic field fluctuations, cannot be reversed.
2.4.3 Ramsey spectroscopy in a magnetic guiding field
In previous experiments we have extensively studied the dephasing mechanisms of the magnetically insensitive |F = 4, mF = 0i ↔ |F = 3, mF = 0i transition of the cesium atoms in our dipole trap [48]. However, the addressing scheme of the quantum register presented in this thesis requires the use of magnetically sensitive Zeeman states in a mag-netic field. Not surprisingly, the greater sensitivity of the qubits to their environment shortens the dephasing times. For a detailed analysis of the limiting effects I first con-sider the dephasing mechanisms of the atoms in a homogeneous guiding field B0 before studying the coherence properties of the quantum register in the inhomogeneous magnetic quadrupole field.
Ramsey spectroscopy, which was introduced in the context of magnetic resonance exper-iments [77, 78], is the standard technique to study dephasing mechanisms. It measures the total transverse dephasing time T2 by free induction decay [79]. In the Bloch vector picture, an atom prepared in state u0 = (0,0,−1) is transferred to the uv plane by a π/2 pulse whose frequency ω is slightly detuned with respect to the atomic frequency ω0, see Figure 2.14. After free precession at the frequency δ = ω−ω0 during the time τR a second π/2 pulse rotates the Bloch vector around the u axis. A projection mea-surement onto the w axis reveals fringes at the frequency δ. The frequency spread of an
2.4 Coherence properties 71
Figure 2.14: Ramsey spectroscopy. Aπ/2 pulse with a frequency detuning δfrom the atomic resonance rotates the Bloch vector fromw=−1 into theuvplane. After a delay timeτRof free precession at the frequencyδ, the phase of the Bloch vector is measured by application of a secondπ/2 pulse and subsequent projection measurement onto the waxis.
atomic ensemble ∆ω0 causes the Ramsey fringes to decay withinT2∗≈1/∆ω0. We will see that T2∗ T20 so that the total transverse dephasing time is governed by inhomogeneous dephasing mechanisms, T2 ≈T2∗.
Experimental parameters
The experimental sequence and the chosen parameters for this measurement are similar to the recording of Rabi oscillations, see Section 1.6.4. Here, instead of varying the duration of the microwave pulse, we record the final population in state |1i as a function of the delay τR in between two π/2 pulses oftπ/2 = 8 µs duration. The microwave frequency is chosen to be off-resonant by δ/2π= 10 kHz which is small enough compared to the Rabi frequency of ΩR/2π = 30 kHz to reasonably remain in the limit of resonant pulses. The applied magnetic guiding field is B0 = 2 G and the trap depth is 100 µK. The resulting data are plotted in Figure 2.15.
Results – Inhomogeneous dephasing
The timescale for the decay of the Ramsey fringes in Figure 2.15 of roughly 300 µs is noticeably shorter than the dephasing time of 4.4 ms which was obtained for the magneti-cally insensitive transition studied in the thesis of Stefan Kuhr [48]. There, the dominating dephasing mechanism was identified to be the inhomogeneous broadening of the atomic resonance frequency due to the energy dependent differential light shift of the individual trapped atoms, see also Section 1.7.3. I will show in the following that the observed de-phasing in Figure 2.15 is caused by the same mechanism. As the vectorial contribution to the light shift of the outer Zeeman transition is much larger than the scalar contribution (see Section 2.3.2), the dephasing times are correspondingly shorter. I therefore adopt the model developed in the thesis of Stefan Kuhr [48] for our case.
Note, however, that this model is a simplified classical model, in which the atomic motion in the trap is neglected. We recently learned that a full quantum mechanical treatment is required, as done in Reference [80], in order to fully account for the observed dephasing.
A first estimate shows that the additional quantum mechanical effects are of the same order of magnitude as the classical dephasing effects considered below [81]. We believe,
Figure 2.15: Decay of Ramsey fringes due to inhomogeneous broadening of the atomic resonance frequency. The data points show the measured population in state |1i as a function of the time delayτRbetween the twoπ/2 pulses. Every data point is averaged over 10 experimental runs with about 5 atoms each. The fit according to Equation (2.21) reveals an inhomogeneous dephasing time ofT2∗= 270±25µs.
though, that these quantum effects are completely reversible and do not affect the spin echo analysis presented in Section 2.4.5. The full quantum mechanical treatment of the dephasing mechanisms in our trap is subject to future investigations.
In the classical model, the distribution of the differential light shifts of the trapped atoms is derived from their Boltzmann energy distribution. The shape of the Ramsey fringes winh(τR) is the Fourier-Cosine transform of this light shift distribution [72], see Equa-tion (1.34):
winh(τR) =α(τR, T2∗) cos [δτR+κ(τR, T2∗)], (2.17) with a time-dependent amplitudeα(τR, T2∗) and phase shiftκ(τR, T2∗):
α(τR, T2∗) = 1 + 0.95 τR
T2∗
2!−3/2
and κ(τR, T2∗) =−3 arctan
0.97τR
T2∗
. (2.18) The inhomogeneous dephasing time T2∗ is defined as the 1/e−time of the amplitude α(τR, T2∗):
α(T2∗)=! e−1 ⇒ T2∗ =p
e2/3−1β= 0.97β , (2.19) with the temperature-dependent coefficient
β= 2U0
kBT δls,max. (2.20)
Again, δls,max denotes the maximum differential light shift for an atom in the bottom of the potential well. These relations hold as long as the atoms are cold enough to be in the
2.4 Coherence properties 73 harmonic region of the trap.
The data in Figure 2.15 are therefore fitted using the following function which is obtained from Equations (2.17) and (2.19):
P1(τR) =A·α(τR, T2∗) cos [δτR+κ(τR, T2∗) +ϕ] +B. (2.21) In addition to the amplitude A and offset B, a phase ϕ is introduced because the Bloch vector also precesses during the application of the two π/2 pulses while the model only considers free precession during the time τR. The resulting fit parameters are listed in Table 2.4.
Fit results
fringe amplitude A 45.1± 2.1 % fringe offset B 46.6± 1.0 % detuning δ/2π 12.5± 0.1 kHz phase offset ϕ 1.15± 0.08 rad dephasing time T2∗ 270± 25 µs
Table 2.4: Fit results for the Ramsey fringes of Figure 2.15.
From temperature measurements in the dipole trap we know that in a trap of depth U0 = 0.1 mK the temperature of the atoms is T = 22±6 µK [54]. Equations (2.19) and (2.20) therefore allow us to infer the differential light shiftδls,max/2π= 5.2±1.4 kHz from the measured dephasing time T2∗. This result is smaller than the spectroscopically measured differential light of|∆Ediff,tot/h|= 8.5±0.5 kHz for U0 = 0.1 mK presented in Section 2.3.2. This indicates that the actual trap depth for this measurement might have been smaller than assumed, for example due to slight misalignment of the trapping laser beams.
The measured detuning is slightly larger than the actually applied detuning ofδ = 10 kHz.
This is probably due to the fact that the Ramsey fringes were not acquired immediately after the spectroscopic calibration of the atomic resonance frequencyω0. The drifts of the atomic resonance frequency of 1 kHz/h increased the detuning between the two measure-ments.
The measured phase offsetϕdue to the Bloch vector precession during the twoπ/2 pulses roughly confirms the pulse durationtπ/2=ϕ/2δ= 7.3±0.5µs.
2.4.4 Spin echo spectroscopy in a magnetic guiding field
Inhomogeneous dephasing can be reversed by applying a π pulse between the two π/2 pulses. The Bloch vectors dephase due to the slightly different precession frequencies of different atoms. If the π pulse is applied at the time τπ after the firstπ/2 pulse, they will rephase at 2τπ, see Figure 2.16. This phenomenon is referred to as a spin echo and was first implemented in magnetic resonance experiments [82]. Nowadays, this technique
Figure 2.16: Spin echo. The dephasing Bloch vectors are rephased by the application of aπpulse between the two Ramseyπ/2 pulses.
has also been applied to atoms captured in dipole traps [17, 83].
To measure the homogeneous transverse dephasing time T20 of the trapped atoms in the magnetic offset field B0 we performed spin echo spectroscopy for various times τπ. As the contrast of the spin echo fringes decreases with increasing τπ we infer T20 from their envelope.
Experimental parameters
The experimental parameters are identical to the ones used in the previous section. The spin echos are recorded in a trap of depth U0 = 100 µK, in a magnetic guiding field of 2 G, and with a microwave detuning ofδ/2π= 10 kHz.
Results
The recorded spin echo signals are shown in Figure 2.17 (a). Each spin echo signal belongs to a separate measurement with fixed τπ. While the Ramsey fringes of Figure 2.15 decay within T2∗= 270 µs, the spin echo contrast decreases on a longer timescale.
The shape of the spin echo signals is very similar to the Ramsey signal [17]. However, for simplicity, I have neglected T2∗ in Equation (2.21) to yield the following fit function:
P1(t) =−A·cos [δ(t−2τπ) +φ] +B . (2.22) Here, the phase shift φ accounts for slow systematic phase drifts during the spin echo sequence. The contrast is defined by:
C = Pmax−Pmin
Pmax+Pmin = A
B . (2.23)
2.4 Coherence properties 75
Figure 2.17: (a) Spin echo fringes in a magnetic guiding field. While the Ramsey signal decays quickly, the spin echo contrast decreases on a longer timescale. Every data point is obtained from 5 experimental runs with about 9 atoms each. The lines are fits according to Equation (2.22). (b) The spin echo contrast as a function of spin echo time 2τπreveals a homogeneous dephasing time ofT20≈3.5 ms.
The result is shown in Figure 2.17 (b) where I plotted the contrast as a function of the spin echo time 2τπ. In analogy to T2∗, we define the homogeneous dephasing time T20 as the 1/e−time of the contrast:
C(2τπ =T20)=! C(0)e−1. (2.24) From the data points I estimate a homogeneous dephasing time ofT20 ≈3.5 ms.
2.4.5 Dephasing mechanisms in a magnetic guiding field
In the following I discuss the mechanisms that lead to the observed dephasing. So far, the detuningδ has been treated as time independent resulting in a rephasing of the precessing Bloch vectors at exactly 2τπ. However, a time-varying detuning δ(t) results in a shift of the spin echo signal if the difference of the accumulated phases ∆φ(τπ) before and after
the rephasing π pulse does not vanish:
∆φ(τπ) = Z 2τπ
τπ
δ(t0)dt0− Z τπ
0
δ(t0)dt0. (2.25)
The time dependence ofδ(t) is caused by, e. g., magnetic field and intensity fluctuations.
Since the acquisition of a spin echo signal requires several repetitions of the experimental sequence, ∆φ(τπ) will be different every time resulting in a decrease of the fringe contrast.
We express the phase difference (2.25) as an average detuning difference ∆δ= ∆φ(τπ)/τπ and assume its probability distribution to be Gaussian:
p(∆δ, τπ) = 1 σ∆δ(τπ)√
2π exp
− (∆δ)2 2σ∆δ(τπ)2
, (2.26)
with mean ∆δ= 0 and varianceσ∆δ(τ)2. The decay of the spin echo contrastC(τπ) is given by an integral similar to the Fourier-Cosine transform of the distribution of fluctuations p(∆δ, τπ) [48]:
C(τπ) =C0 Z ∞
−∞−cos (∆δ τπ) p(∆δ, τπ)d∆δ. (2.27) Performing the integration yields:
C(τπ) =C0 exp
−1
2 τπ2σ∆δ(τπ)2
. (2.28)
Magnetic field fluctuations
The observed homogeneous dephasing is most likely caused by magnetic field fluctuations of the guiding field. At the time when we recorded the Ramsey and spin echo measurements presented so far, we only possessed a noisier current supply (Elektro-Automatik, EA-PS 3016-10) to provide the guiding field along the trap axis. Since its relative current stability of Irms/I0 = 10−3 was insufficient to measure spin echo fringes, we used a low-pass filter with a time-constant of 10 ms to increase the magnetic field stability in the frequency range above 100 Hz. This improvement allowed us to measure the spin echo fringes presented in Figure 2.17. According to Equations (2.24) and (2.28) the decay of the contrast yields a standard deviation for the average detuning difference of σ∆δ(T20) = 2π·130 Hz, see Table 2.5.
The minimum relative current stability inferred from this result is therefore:
Irms
I0 = Brms
B0 = σ∆δ(T20)
δ0 = 2.5·10−5. (2.29)
Fluctuations of the magnetic field caused by other devices in the vicinity of the atom trap remain to be investigated. Since the measured dephasing time already demonstrates that the magnetic field fluctuations are smaller thanBrms= 50µG, it might be sensible to use the high sensitivity of the qubit transition itself as a probe for measuring and improving the magnetic field stability.
2.4 Coherence properties 77 Beam pointing instability of the dipole trap laser
In previous work [72] we investigated the pointing stability of the two dipole trap laser beams. The fluctuating optical contrast of the standing wave due to beam pointing insta-bilities was identified as the dominant mechanism for the homogeneous dephasing time of T20= 68 ms of the magnetic field insensitive |F = 3, mF= 0i ↔ |F = 4, mF= 0i transi-tion.
The qubit transition investigated here is more sensitive to this effect due to non-vanishing contributions of circular light polarizations, see Section 2.3.2. Taking also into account the circular light shift, we expect a homogenous dephasing time of T20 = 12 ms.
Intensity fluctuations
Similarly to the beam pointing instabilities, intensity fluctuations of the dipole trap laser contribute to homogeneous dephasing. The resulting fluctuations of the trap depth cause fluctuations of the differential light shift and thus of the detuning between the microwave frequency and the atomic resonance frequency. Transferring the results from previous investigations [48] to the present case, the expected homogeneous dephasing time due to intensity fluctuations amounts to T20 = 23 ms.
Further dephasing mechanisms
Other dephasing mechanisms such as elastic collisions, heating, and fluctuations of the microwave power and pulse duration have been investigated [48] and found to be several orders of magnitude weaker than the dominating dephasing effects discussed above.
An additional source of dephasing is the time-dependent differential light shift due to the oscillatory motion of the atoms in the trap. This effect is fully analyzed and discussed for the case of the modulation of the atomic resonance frequency in the magnetic field gradient
Homogeneous dephasing time
Fluctuation amplitude T20 σ∆δ(T20)
2π
measured 3.5 ms 130 Hz
Dephasing mechanisms
magnetic field fluctuations ≥3.5 ms ≤130 Hz
beam pointing instability 12 ms 37 Hz
intensity fluctuations 23 ms 19 Hz
Table 2.5: Summary of the relevant irreversible dephasing mechanisms in a magnetic guiding field.
in Section 2.4.7. It causes a periodic decay and revival of the spin echo contrast at twice the radial oscillation frequency of the trapped atoms. For the value of the differential light shiftδls,max/2π = 8.5 kHz in a trap ofU0 = 100µK, as measured in Section 2.3.2, I would expect a periodic decay of the contrast by 22 %. However, since this effect is not observed in the measurement, I conclude that the differential light shift is δls,max/2π = 5 kHz as derived from the Ramsey fringes of Section 2.4.3 or even smaller. For these values, the periodic reduction of the contrast amounts to less than 10 %.
Finally, population relaxation of the two ground states, characterized by the decay time T1, which is caused by scattering of photons from the dipole trap laser, was also measured in our group [41]. In a trap of depth U0 = 100 µK the resulting population decay time amounts to T1= 86 s and is completely negligible in this context.
I have summarized the measured and the expected homogenous dephasing times for the relevant dephasing mechanisms in Table 2.5, along with the fluctuation amplitude σ∆δ at the respective timeT20.
2.4.6 Spin echo spectroscopy in a magnetic field gradient
After analysis of the dephasing mechanisms of trapped atoms in a homogeneous magnetic guiding field, I now study the dephasing time in the inhomogeneous magnetic field gradi-ent (2.1) because it determines the coherence time of the quantum register. Adding the quadrupole field to the guiding field gives rise to additional dephasing mechanisms. The coupling of external and internal degrees of freedom via the field gradient has the side effect that position fluctuations of the atoms translate into fluctuations of the qubit frequency.
We will see in Section 2.4.7 that the radial oscillations of the atoms in the trap limit the coherence time of the quantum register. On a longer timescale, axial position fluctuations of the standing wave dipole trap constitute an additional dephasing mechanism.
A spin echo measurement of the quantum register is significantly more complicated and time consuming than the presented measurements in a magnetic guiding field, because it has to be performed on single atoms. After transfer from the MOT into the dipole trap, the atoms are distributed along the trap axis over at least 20−30 µm. At the gradient of B0 = −1.5 mG/µm, this corresponds to a frequency spread of up to 110 kHz which is much larger than the Rabi frequency. Ramsey or spin echo spectroscopy on such an atomic cloud will therefore fail. For this reason, we perform a spin echo measurement on single atoms only, combining our addressing technique presented in Section 2.3.1 with the spin echo technique demonstrated in Section 2.4.4.
Experimental sequence
A schematic overview of the experimental sequence is shown in Figure 2.18. We start by loading atoms into the MOT. The loading time has been optimized such that the average atom number is one. After transfer into the dipole trap we acquire an image of the atom and determine its position and resonance frequency. The magnetic gradient field B(r), see Equation (2.1), is switched on, withB0= 4 G and B0 =−1.5 mG/µm. After lowering
2.4 Coherence properties 79
Figure 2.18: Experimental sequence for the spin echo measurement of the quantum register. After transfer from the MOT into the dipole trap, we image the atom with the ICCD. In the lowered trap, the atom is initialized by optical pumping and subjected to the resonant spin echo pulse sequence. State-selective detection is performed by application of the push-out laser and subsequent fluorescence detection in the MOT.
the trap depth to 100µK and initializing the atom in state|0i, we apply the first Ramsey π/2 pulse at t=t0. For simplicity, the pulse shape is rectangular, with a pulse duration oftπ/2 = 8µs. The microwave frequency is tuned in resonance with the atomic frequency, δ= 0, as discussed below. At t=t0+τπ, we apply the spin echo π pulse. Since complete rephasing is expected at t =t0+ 2τπ, we then apply the final π/2 pulse and detect the state of the atom by subjecting it to the push-out laser and by subsequently transferring it into the MOT.
This sequence is repeated 100 times which takes a total acquisition time of five minutes.
We post-select all experimental runs with exactly one atom present so that the resulting data point contains roughly 40 single atom events. The measurement of an entire spin echo signal as demonstrated in Section 2.4.4 is impossible due to the drifts of the atomic resonance frequency caused by the time-varying differential light shift over the long data acquisition time. We therefore infer the spin echo contrast by simply measuring the minimumPmin of the spin echo signal. Since the resonant transfer efficiency including all experimental imperfections amounts to 100+0.0−5.1 %, we set Pmax = 1−Pmin. Thus, the contrast is equal to:
C = Pmax−Pmin
Pmax+Pmin = 1−2Pmin. (2.30) Although the spin echo minimum is expected at t = t0 + 2τπ, homogeneous systematic frequency drifts lead to a small shift of this minimum. This effect was also observed in the spin echo signals recorded in the magnetic guiding field, see Figure 2.17, and was formally accounted for by an additional phase shift φ in Equation (2.22). In order to measure Pmin more precisely, we therefore acquire up to five data points around the minimum of the echo signal at t = t0+ 2τπ ±ε, with ε being varied in steps of 30 µs. In order to minimize the number of data points required to determinePmin reasonably well, we apply